Introduction to Machine Learning - CS725 Instructor: Prof. Ganesh Ramakrishnan Lecture 4 - Linear Regression - Probabilistic Interpretation and Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Recap: Linear Regression is not Naively Linear Need to determine w for the linear function f ( x , w ) = ∑ n i =1 w i φ i ( x j ) = Φw which minimizes our error function E ( f ( x , w ) , D ) Owing to basis function φ , “Linear Regression” is linear in w but NOT in x (which could be arbitrarily non-linear)! φ 1 ( x 1 ) φ 2 ( x 1 ) ...... φ p ( x 1 ) . Φ = (1) . φ 1 ( x m ) φ 2 ( x m ) φ n ( x m ) ...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Recap: Linear Regression is not Naively Linear Need to determine w for the linear function f ( x , w ) = ∑ n i =1 w i φ i ( x j ) = Φw which minimizes our error function E ( f ( x , w ) , D ) Owing to basis function φ , “Linear Regression” is linear in w but NOT in x (which could be arbitrarily non-linear)! φ 1 ( x 1 ) φ 2 ( x 1 ) ...... φ p ( x 1 ) . Φ = (1) . φ 1 ( x m ) φ 2 ( x m ) φ n ( x m ) ...... Least Squares error and corresponding estimates: E ∗ = min ( ) w T Φ T Φw − 2y T Φw + y T y w E ( w , D ) = min (2) w ( n ) 2 m w ∗ = arg min ∑ ∑ E ( w , D ) = arg min w i φ i ( x j ) − y j w w . . . . . . . . . . . . . . . . . . . . j = 1 i = 1 . . . . . . . . . . . . . . . . . . . . (3)
Recap: Geometric Interpretation of Least Square Solution Let y ∗ be a solution in the column space of Φ The least squares solution is such that the distance between y ∗ and y is minimized Therefore, the line joining y ∗ to y should be orthogonal to the column space of Φ ⇒ w = ( Φ T Φ ) − 1 Φ T y (4) Here Φ T Φ is invertible only if Φ has full column rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Building on questions on Least Squares Linear Regression 1 Is there a probabilistic interpretation? Gaussian Error, Maximum Likelihood Estimate 2 Addressing overfitting Bayesian and Maximum Aposteriori Estimates, Regularization 3 How to minimize the resultant and more complex error functions? Level Curves and Surfaces, Gradient Vector, Directional Derivative, Gradient Descent Algorithm, Convexity, Necessary and Sufficient Conditions for Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Probabilistic Modeling of Linear Regression Linear Model: Y is a linear function of φ ( x ), subject to a random noise variable ε which we believe is ‘mostly’ bounded by some threshold σ : Y = w T φ ( x ) + ε ε ∼ N (0 , σ 2 ) Motivation: N ( µ, σ 2 ), has maximum entropy among all real-valued distributions with a specified variance σ 2 3 − σ rule: About 68% of values drawn from N ( µ, σ 2 ) are within one standard deviation σ away from the mean µ ; about 95% of the values lie within 2 σ ; and about 99.7% are within 3 σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Figure 1: 3 − σ rule: About 68% of values drawn from N ( µ, σ 2 ) are within one standard deviation σ away from the mean µ ; about 95% of the values lie within 2 σ ; and about 99.7% are within 3 σ . Source: https://en.wikipedia.org/wiki/Normal_distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Probabilistic Modeling of Linear Regression Linear Model: Y is a linear function of φ ( x ), subject to a random noise variable ε which we believe is ‘mostly’ around some threshold σ : Y = w T φ ( x ) + ε ε ∼ N (0 , σ 2 ) This allows for the Probabilistic model P ( y j | w , x j , σ 2 ) = N ( w T φ ( x j ) , σ 2 ) m ∏ P ( y | w , x j , σ 2 ) = P ( y j | w , x j , σ 2 ) j =1 Another motivation: E [ Y ( w , x j )] = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Probabilistic Modeling of Linear Regression Linear Model: Y is a linear function of φ ( x ), subject to a random noise variable ε which we believe is ‘mostly’ around some threshold σ : Y = w T φ ( x ) + ε ε ∼ N (0 , σ 2 ) This allows for the Probabilistic model P ( y j | w , x j , σ 2 ) = N ( w T φ ( x j ) , σ 2 ) m ∏ P ( y | w , x j , σ 2 ) = P ( y j | w , x j , σ 2 ) j =1 Another motivation: E [ Y ( w , x j )] = w T φ ( x j ) = w T 0 + w T 1 φ 1 ( x j ) + ... + w T n φ n ( x j ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Estimating w : Maximum Likelihood If ϵ ∼ N (0 , σ 2 ) and y = w T φ ( x ) + ϵ where w , φ ( x ) ∈ R m then, given dataset D , find the most likely ˆ w ML ( ( y j − w T φ ( x j )) 2 1 ) Recall: Pr( y j | x j , w ) = √ 2 πσ 2 exp 2 σ 2 From Probability of data to Likelihood of parameters : Pr( D| w ) = Pr( y | x , w ) = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Estimating w : Maximum Likelihood If ϵ ∼ N (0 , σ 2 ) and y = w T φ ( x ) + ϵ where w , φ ( x ) ∈ R m then, given dataset D , find the most likely ˆ w ML ( ( y j − w T φ ( x j )) 2 1 ) Recall: Pr( y j | x j , w ) = √ 2 πσ 2 exp 2 σ 2 From Probability of data to Likelihood of parameters : Pr( D| w ) = Pr( y | x , w ) = m m ( ( y j − w T φ ( x j )) 2 ) 1 ∏ ∏ Pr( y j | x j , w ) = √ 2 πσ 2 exp 2 σ 2 j =1 j =1 Maximum Likelihood Estimate w ML = argmax ˆ Pr( D| w ) = Pr( y | x , w ) = L ( w |D ) w . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Optimization Trick Optimization Trick: Optimal point is invariant under monotonically increasing transformation (such as log ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Optimization Trick Optimization Trick: Optimal point is invariant under monotonically increasing transformation (such as log ) log L ( w |D ) = LL ( w |D ) = m 1 − m ∑ 2 ln (2 πσ 2 ) − ( w T φ ( x j ) − y j ) 2 2 σ 2 j =1 For a fixed σ 2 w ML = ˆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Optimization Trick Optimization Trick: Optimal point is invariant under monotonically increasing transformation (such as log ) log L ( w |D ) = LL ( w |D ) = m 1 − m ∑ 2 ln (2 πσ 2 ) − ( w T φ ( x j ) − y j ) 2 2 σ 2 j =1 For a fixed σ 2 w ML = argmax LL ( y 1 ... y m | x 1 . . . x m , w , σ 2 ) ˆ m ∑ ( w T φ ( x j ) − y j ) 2 = argmin j =1 Note that this is same as the Least square solution!! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Building on questions on Least Squares Linear Regression 1 Is there a probabilistic interpretation? Gaussian Error, Maximum Likelihood Estimate 2 Addressing overfitting Bayesian and Maximum Aposteriori Estimates, Regularization 3 How to minimize the resultant and more complex error functions? Level Curves and Surfaces, Gradient Vector, Directional Derivative, Gradient Descent Algorithm, Convexity, Necessary and Sufficient Conditions for Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Redundant Φ and Overfitting Figure 2: Root Mean Squared (RMS) errors on sample train and test datasets as a function of the degree t of the polynomial being fit Too many bends (t=9 onwards) in curve ≡ high values of some w ′ i s . Try plotting values of w i ’s using applet at http://mste.illinois.edu/users/exner/java.f/leastsquares/#simulation Train and test errors differ significantly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
X^0 * 0.13252679175596802 X^1 * 6.836159339696569 X^2 * -10.198794083500966 X^3 * 8.298738913209064 X^4 * -3.766949862252123 X^5 * 1.0274981119277349 X^6 * -0.17218031550131038 X^7 * 0.017340835860554016 X^8 * -9.623065771393043E-4 X^9 * 2.2595409656184083E-5 X^0 * -1.4218758581602278 X^1 * 14.756472312089675 X^2 * -24.299789484296475 X^3 * 20.63606795357865 X^4 * -9.934453145766518 X^5 * 2.8975181063446613
Bayesian Linear Regression The Bayesian interpretation of probabilistic estimation is a logical extension that enables reasoning with uncertainty but in the light of some background belief Bayesian linear regression : A Bayesian alternative to Maximum Likelihood least squares regression Continue with Normally distributed errors Model the w using a prior distribution and use the posterior over w as the result Intuitive Prior: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Recommend
More recommend
